How many different words can be formed by using all the letters of the word 'ALLAHABAD'?

Explanation

There are 9 letters in the word 'ALLAHABAD' in which 'A' occur 4 times. 'L' occur 2 times and rest are different.
`:.` Total words `= (9!)/(4!.2!) = 7560`
No. of even places = 4
Vowels = A,A,A, A
No. of ways to fill even places by 4 vowels `= (4!)/(4!) = 1`
No. of ways to fill remaining 5 odd positions by remaining 5 letters
`= (5!)/(2!) = 60 ( :' L` occur 2 times)
Now from multiplication theorem,
No. of words in which vowels occupy even positions `= 1 xx 60 = 60`.